Oblique asymptotes - What am I doing wrong?

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SUMMARY

The discussion centers on finding the oblique asymptote of the function f(x) = (2x^3 + 4x^2 - x + 1) / (-x^2 - x + 2). The correct oblique asymptote is determined to be f(x) = -2x - 2, verified through polynomial long division. The confusion arises when attempting to manipulate the function by multiplying by (1/x^2)/(1/x^2), leading to an incorrect limit of -2x - 4 instead of the established asymptote. The error is attributed to the contribution of the denominator affecting the constant term in the limit calculation.

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f(x) = (2x^3 + 4x^2 - x + 1) / (-x^2 - x + 2)

The limit of this function as x approaches infinity is the oblique asymptote f(x) = -2x - 2
This can be verified by performing long division with the two polynomials to get:

f(x) = -2x -2 + (x +5)/(-x^2 - x + 2)
as x -> infinity, the term (x +5)/(-x^2 - x + 2) -> zero

Now my question is why does the following not work:

f(x) = (2x^3 + 4x^2 - x + 1)/(-x^2 - x + 2) * (1/x^2)/(1/x^2)

f(x) = (2x + 4 - 1/x + 1/x^2) / (-1 - 1/x + 2/x^2)

lim f(x) as x -> infinity should then be -2x - 4 because all other terms approach zero as x gets larger and larger right?

why is this wrong? I am so confused... I thought the answers should have been the same!
 
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The denominator in your expression makes a contribution to the constant term.
1/(-1-1/x)~-1+1/x. Therefore f(x)~(2x+4)(-1+1/x)~(-2x-4)+(2+4/x)~-2x-2
 

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